Math#6 Equations

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Table of contents

## Contents

* 1 Latex equations
* 2 Solving equations
* 3 Solving linear, quadratic and cubic equations
* 4 Polynomial solving (symbolic)
* 5 Algebraic equation systems
* 6 Solving linear systems
* 7 Solving non-linear systems
* 8 Solving differential equations

# Latex equations

Equation is a statement of an equality containing one or more variables. You can create PNG images with equations using the Software:LaTeX syntax using the class  jhplot.HLatexEq. These images can be included on the web page or presentations. Here is a small code that shows how to make a PNG image

from jhplot import *
eq="\int^{100}_{i=0} F(x) dx" # use LaTeX syntax for this equation
image="/tmp/equation.png"     # output file with PNG image of this equation
q1=HLatexEq(eq, 32)           # create PNG image from LaTeX using the font size 32
q1.export(image)              # making the image
print "Created :",image
IView(image)                  #  View the created PNG image

# Solving equations

Numerous Java packages can be used to solve linear, non-linear and differential equations.

# Solving linear, quadratic and cubic equations

To solve linear, quadratic and cubic equations, use the  jhplot.math.Numeric package. In general, only real solutions are considered.

from jhplot.math.Numeric import *
a=solveLinear(1,2) # solves ax+b=0. a=1, b=2
print a
b= solveQuadratic(1, 2, -1) # roots of the quadratic equation
print b
b= solveCubic(1, 2, 4,2)
print b
c=solveQuartic(1,-2, 3, 4, -2)
print b

# Polynomial solving (symbolic)

For solving polynomial equations, one can use the symbolic calculation engine.

from jhplot.math import *
from jhplot import *
j=Symbolic("jscl") # using jscl engine
j.expand("solve(c+b*x+a*x^2,x)") # answer: root[0](c, b, a)

# Algebraic equation systems

One solve algebraic equation systems of any degree, with several indeterminates, by computing the Groebner bases of polynomial ideals. For example, let this system for the indeterminates x, y:

x^2 + y^2 = 4
x*y = 1

We use:

j.expand("groebner({x^2 + y^2 - 4, x*y - 1}, {x, y})")

The returned output is:

{1-4*x^2+x^4, 4*x-x^3-y}

which allows to find x, then y from x. This operation doesn't calculate the roots, it just writes the equation. For example, it wouldn't give "a = 4/5" but "5*a-4" ("= 0" implied). Groebner basis computation is explained in more details below:

# Solving linear systems

Consider a linear systems of equations of the form AX=B. For example, consider

2x + 3y - 2z =  1
x + 7y + 6x = -2
4x - 3y - 5z =  1

We will solve this using  org.apache.commons.math3.linear.DecompositionSolver package:

from org.apache.commons.math3.linear import *

# get the coefficient matrix A using LU decomposition
coeff= Array2DRowRealMatrix([[2,3,-2],[-1,7,6],[4,-3,-5]])
solver =LUDecompositionImpl(coeff).getSolver()
#  use solve(RealVector) to solve the system
constants = ArrayRealVector([1, -2, 1 ])
solution = solver.solve(constants);
print "Solution: x=",solution.getEntry(0), "y=",solution.getEntry(1),"z=",solution.getEntry(2)

The execution of this code prints:

Solution: x= -0.369863013699 y= 0.178082191781 z= -0.602739726027

Read more for different types of decomposition here. Please read more in the Linear Algebra section.

# Solving non-linear systems

You can solve non-linear equations using the jMathLab symbolic kernel as explained in Section JMathLab Equations.

# Solving differential equations

under construction

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